Peter McMullen | |
---|---|

Born | 11 May 1942 |

Nationality | British |

Alma mater | Trinity College, Cambridge |

Known for | Upper bound theorem, McMullen problem |

Scientific career | |

Fields | Discrete geometry |

Institutions | Western Washington University (1968–1969) University College London |

**Peter McMullen** (born 11 May 1942)^{ [1] } is a British mathematician, a professor emeritus of mathematics at University College London.^{ [2] }

McMullen earned bachelor's and master's degrees from Trinity College, Cambridge, and studied at the University of Birmingham, where he received his doctorate in 1968.^{ [3] } and taught at Western Washington University from 1968 to 1969.^{ [4] } In 1978 he earned his Doctor of Science at University College London where he still works as a professor emeritus. In 2006 he was accepted as a corresponding member of the Austrian Academy of Sciences.^{ [5] }

McMullen is known for his work in polyhedral combinatorics and discrete geometry, and in particular for proving what was then called the upper bound conjecture and now is the upper bound theorem. This result states that cyclic polytopes have the maximum possible number of faces among all polytopes with a given dimension and number of vertices.^{ [6] } McMullen also formulated the g-conjecture, later the g-theorem of Louis Billera, Carl W. Lee, and Richard P. Stanley, characterizing the *f*-vectors of simplicial spheres.^{ [7] }

The McMullen problem is an unsolved question in discrete geometry named after McMullen, concerning the number of points in general position for which a projective transformation into convex position can be guaranteed to exist. It was credited to a private communication from McMullen in a 1972 paper by David G. Larman.^{ [8] }

He is also known for his 1960s drawing, by hand, of a 2-dimensional representation of the Gosset polytope 4_{21}, the vertices of which form the vectors of the E8 root system.^{ [9] }

McMullen was invited to speak at the 1974 International Congress of Mathematicians in Vancouver; his contribution there had the title *Metrical and combinatorial properties of convex polytopes*.^{ [10] }

He was elected as a foreign member of the Austrian Academy of Sciences in 2006.^{ [11] } In 2012 he became an inaugural fellow of the American Mathematical Society.^{ [12] }

- Research papers

- McMullen, P. (1970), "The maximum numbers of faces of a convex polytope",
*Mathematika*,**17**(2): 179–184, doi:10.1112/s0025579300002850, MR 0283691, S2CID 122025424 . - —— (1975), "Non-linear angle-sum relations for polyhedral cones and polytopes",
*Mathematical Proceedings of the Cambridge Philosophical Society*,**78**(2): 247–261, Bibcode:1975MPCPS..78..247M, doi:10.1017/s0305004100051665, MR 0394436, S2CID 63778391 . - —— (1993), "On simple polytopes",
*Inventiones Mathematicae*,**113**(2): 419–444, Bibcode:1993InMat.113..419M, doi:10.1007/BF01244313, MR 1228132, S2CID 122228607 .

- Survey articles

- ——; Schneider, Rolf (1983), "Valuations on convex bodies",
*Convexity and its applications*, Basel: Birkhäuser, pp. 170–247, MR 0731112 . Updated as "Valuations and dissections" (by McMullen alone) in*Handbook of convex geometry*(1993), MR 1243000.

- Books

- ——; Shephard, Geoffrey C. (1971),
*Convex Polytopes and the Upper Bound Conjecture*, Cambridge University Press. - ——; Schulte, Egon (2002),
*Abstract regular polytopes*, Encyclopedia of Mathematics and its Applications, vol. 92, Cambridge: Cambridge University Press, doi:10.1017/CBO9780511546686, ISBN 0-521-81496-0, MR 1965665, S2CID 115688843 .

In geometry, a **polyhedron** is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.

In geometry, the **convex hull** or **convex envelope** or **convex closure** of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.

**Discrete geometry** and **combinatorial geometry** are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

In geometry, a **net** of a polyhedron is an arrangement of non-overlapping edge-joined polygons in the plane which can be folded to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard.

A **convex polytope** is a special case of a polytope, having the additional property that it is also a convex set contained in the -dimensional Euclidean space . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary.

In mathematics, the "**happy ending problem**" is the following statement:

In geometry, a **vertex**, often denoted by letters such as , , , , is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices.

**Combinatorial commutative algebra** is a relatively new, rapidly developing mathematical discipline. As the name implies, it lies at the intersection of two more established fields, commutative algebra and combinatorics, and frequently uses methods of one to address problems arising in the other. Less obviously, polyhedral geometry plays a significant role.

In algebraic combinatorics, the ** h-vector** of a simplicial polytope is a fundamental invariant of the polytope which encodes the number of faces of different dimensions and allows one to express the Dehn–Sommerville equations in a particularly simple form. A characterization of the set of

**Polyhedral combinatorics** is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes.

In polyhedral combinatorics, a branch of mathematics, **Steinitz's theorem** is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs.

In geometry and combinatorics, a **simplicial**** d-sphere** is a simplicial complex homeomorphic to the

**Louis Joseph Billera** is a Professor of Mathematics at Cornell University.

In geometry and polyhedral combinatorics, a **k-neighborly polytope** is a convex polytope in which every set of k or fewer vertices forms a face. For instance, a 2-neighborly polytope is a polytope in which every pair of vertices is connected by an edge, forming a complete graph. 2-neighborly polytopes with more than four vertices may exist only in spaces of four or more dimensions, and in general a k-neighborly polytope requires a dimension of 2*k* or more. A d-simplex is d-neighborly. A polytope is said to be **neighborly**, without specifying k, if it is k-neighborly for *k* = ⌊*d*⁄2⌋. If we exclude simplices, this is the maximum possible k: in fact, every polytope that is k-neighborly for some *k* ≥ 1 + ⌊*d*⁄2⌋ is a simplex.

In mathematics, a **cyclic polytope**, denoted *C*(*n*,*d*), is a convex polytope formed as a convex hull of *n* distinct points on a rational normal curve in **R**^{d}, where *n* is greater than *d*. These polytopes were studied by Constantin Carathéodory, David Gale, Theodore Motzkin, Victor Klee, and others. They play an important role in polyhedral combinatorics: according to the upper bound theorem, proved by Peter McMullen and Richard Stanley, the boundary *Δ*(*n*,*d*) of the cyclic polytope *C*(*n*,*d*) maximizes the number *f*_{i} of *i*-dimensional faces among all simplicial spheres of dimension *d* − 1 with *n* vertices.

**Geoffrey Colin Shephard** is a mathematician who works on convex geometry and reflection groups. He asked Shephard's problem on the volumes of projected convex bodies, posed another problem on polyhedral nets, proved the Shephard–Todd theorem in invariant theory of finite groups, began the study of complex polytopes, and classified the complex reflection groups.

The **McMullen problem** is an open problem in discrete geometry named after Peter McMullen.

In geometry, **Kalai's 3 ^{d} conjecture** is a conjecture on the polyhedral combinatorics of centrally symmetric polytopes, made by Gil Kalai in 1989. It states that every

In mathematics, the **upper bound theorem** states that cyclic polytopes have the largest possible number of faces among all convex polytopes with a given dimension and number of vertices. It is one of the central results of polyhedral combinatorics.

* Convex Polytopes* is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional convex polyhedra. It was written by Branko Grünbaum, with contributions from Victor Klee, Micha Perles, and G. C. Shephard, and published in 1967 by John Wiley & Sons. It went out of print in 1970. A second edition, prepared with the assistance of Volker Kaibel, Victor Klee, and Günter M. Ziegler, was published by Springer-Verlag in 2003, as volume 221 of their book series Graduate Texts in Mathematics.

- ↑ Peter McMullen, Peter M. Gruber, retrieved 2013-11-03.
- ↑ UCL IRIS information system, accessed 2013-11-05.
- ↑ McMullen, Peter; Schulte, Egon (12 December 2002),
*Abstract and regular polytopes*, ISBN 9780521814966 , retrieved 2022-05-11 - ↑ Peter McMullen Collection, 1967-1968, Special Collections, Wilson Library, Western Washington University, retrieved from worldcat.org 2013-11-03.
- ↑ "Austrian Academy of Sciences: Peter McMullen" . Retrieved 2022-05-11.
- ↑ Ziegler, Günter M. (1995),
*Lectures on Polytopes*, Graduate Texts in Mathematics, vol. 152, Springer, p. 254, ISBN 9780387943657,Finally, in 1970 McMullen gave a complete proof of the upper-bound conjecture – since then it has been known as the upper bound theorem. McMullen's proof is amazingly simple and elegant, combining to key tools: shellability and

*h*-vectors. - ↑ Gruber, Peter M. (2007),
*Convex and discrete geometry*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 336, Berlin: Springer, p. 265, ISBN 978-3-540-71132-2, MR 2335496,The problem of characterizing the

.*f*-vectors of convex polytopes is ... far from a solution, but there are important contributions towards it. For simplicial convex polytopes a characterization was proposed by McMullen in the form of his celebrated*g*-conjecture. The*g*-conjecture was proved by Billera and Lee and Stanley - ↑ Larman, D. G. (1972), "On sets projectively equivalent to the vertices of a convex polytope",
*The Bulletin of the London Mathematical Society*,**4**: 6–12, doi:10.1112/blms/4.1.6, MR 0307040 - ↑ "A picture of the E8 root system". American Institute of Mathematics. Retrieved 2022-05-11.
- ↑ ICM 1974 proceedings Archived 2017-12-04 at the Wayback Machine .
- ↑ Awards, Appointments, Elections & Honours, University College London, June 2006, retrieved 2013-11-03.
- ↑ List of AMS fellows, retrieved 2013-11-03.

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